The Chern character of a transversally elliptic symbol and the equivariant index
Let \(G\) be a compact Lie group acting on a compact manifold \(M\) with the cotangent bundle \(T^* M\), \(T^*_G M\) be the union of the spaces \((T^*_G M)_x\), \(x\in M\), where \((T^*_G M)_x \subset (T^*_x M)\) is the orthogonal of the tangent space at \(x\) to the orbit \(G(x)\). The \(G\)-equivariant index of the \(G\)-transversally elliptic pseudodifferential operator \(P\) on \(M\) is the generalized function on \(G\) defined by \[ \text{index}^G(P)(g):= \text{Tr}(g, \text{Ker } P)- \text{Tr}(g, \text{Ker } P^*). \] The principal symbol \(\sigma(P)\) defines an element \([\sigma(P)]\) of \(K_G(T^*_G M)\). Moreover, the \(G\)-equivariant index of \(P\) depends only on the class \([\sigma(P)]\in K_G(T^*_G M)\). Thus, the \(G\)-equivariant index induces a homomorphism of \(R(G)\)-modules \[ \text{index}^{G, M}_a: K_G(T^*_G M)\to C^{- \infty}(G)^G \] which is called the analytical index. The purpose of this article is to define the cohomological index \[ \text{index}^{G, M}_c: K_G (T^*_G M) \to C^{- \infty} (G)^G, \] by means of the bouquet integral of a certain family of equivariant differential forms on the cotangent bundle \(T^* M\). The purpose of the next article [the authors, Invent. Math. 124, 51-101 (1996)] is to prove that these two maps -- the analytical and the cohomological index -- coincide.
compact Lie group acting on a compact manifold
\(G\)-transversally elliptic pseudodifferential operator
scientific article; zbMATH DE number 866084